Structure of Asymptotic Space of the Lobachevsky Plane

摘要

The notion of asymptotic space, or asymptotic cone, has been introduced by M.Gromov (5) and is intended to capture the structure of metric spaces 'at infinity'. Let X = (X,d) be a metric space with the set of points X and the distance function d(x,y). Assume that the space X is unbounded, or has infinite diameter; this means that (sup x,y)d(x,y) = infinity. In the original definition it is also required that X is a geodesic space, i.e. for every two points x,y = X there is a isometric embedding i : (O,a) -> X, such that i(O) = x, i(a) = y, a = d(x,y); but we shall not impose this restriction, having in mind different examples of metric spaces.